(N, ε)-pseudospectra of bounded linear operators on ultrametric Banach spaces

Abstract

In this paper, we prove that the essential pseudospectrum of bounded linear operator pencils is invariant under perturbation of completely continuous linear operators on ultrametric Banach spaces over a spherically complete field K and we establish a characterization of the essential pseudospectrum of a bounded linear operator pencils by means of the spectra of all perturbed completely continuous operators. Furthermore, we introduce and study the notion of (n,ε)-pseudospectrum of bounded linear operators and the concept of (n,ε)-pseudospectrum of bounded linear operator pencils on ultrametric Banach spaces. We establish some results about them. Finally, several examples are provided.